CN112488486A - Multi-criterion decision method based on zero sum game - Google Patents

Multi-criterion decision method based on zero sum game Download PDF

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CN112488486A
CN112488486A CN202011338049.XA CN202011338049A CN112488486A CN 112488486 A CN112488486 A CN 112488486A CN 202011338049 A CN202011338049 A CN 202011338049A CN 112488486 A CN112488486 A CN 112488486A
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陈静
唐傲天
刘震
徐森
崔晓凡
胡金旭
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Abstract

The invention discloses a multi-criterion decision method based on a zero sum game, which comprises the following steps: step one, determining an optimized Pareto solution set and n objective functions according to a multi-objective optimization problem; secondly, determining an optimal solution of the objective function; step three, constructing a zero sum game model, converting the zero sum game model into a linear programming problem, and calculating to obtain a mixing strategy of a first participant; and step four, determining a corresponding objective function according to a mixing strategy of the first participant, and mapping the objective function to the optimized Pareto solution set in a coordinate system to obtain the optimal multi-criterion decision. The invention has fast solving speed and high precision.

Description

Multi-criterion decision method based on zero sum game
Technical Field
The invention relates to the technical field of multi-criterion decision, in particular to a multi-criterion decision method based on zero sum game.
Background
The current common multi-criterion decision-making methods are divided into subjective methods and objective methods. For the subjective method, a decision maker gives the weight of each decision target attribute according to investigation and prior and performs decision selection. For an objective method, such as an information entropy method, the concept of entropy in an information theory is used for multi-criterion decision making. When solving a multi-criteria decision problem, a decision needs to be made in the face of multiple conflicting attributes or objectives, the purpose of the decision being to reconcile conflicts between multiple attributes.
The game theory is a scientific theory for researching and resolving conflict. Through investigation on the existing multi-criterion decision-making technology, it is found that the multi-criterion decision-making is not solved from the perspective of game theory.
Disclosure of Invention
The invention designs and develops a multi-criterion decision method based on a zero-sum game, determines a Pareto solution set to reconstruct and solve a zero-sum game model according to a multi-objective optimization problem, namely the solution of the multi-criterion decision, and quickly and accurately finds the optimal multi-criterion decision.
The technical scheme provided by the invention is as follows:
a multi-criterion decision method based on zero sum game comprises the following steps:
the method comprises the following steps of firstly, extracting a plurality of sample points and collecting corresponding response values according to a multi-objective optimization problem, and establishing an approximate model;
determining an optimized Pareto solution set and n objective functions according to the approximate model;
step three, determining the optimal solution of the n objective functions;
step four, constructing a zero and game model:
Figure BDA0002797834060000021
Figure BDA0002797834060000022
Figure BDA0002797834060000023
in the formula, thetaiFor the first participant's mixing strategy, i ═ 1,2, …, n,
Figure BDA0002797834060000024
a blending policy for the second participant, j ═ 1,2, …, n, G, the solution to the first participant's payment matrix;
and step five, determining a corresponding objective function according to a mixing strategy of the first participant, and mapping the objective function to the optimized Pareto solution set in a coordinate system to obtain the optimal multi-criterion decision.
Preferably, the zero-sum game includes: the first participant and the second participant act as gaming participants.
Preferably, the policy set of the first participant is an objective function fi∈{f1,f2,…,fnThe strategy set of the second participant is a set x of optimal solutions of the objective functioni∈{x1,x2,…,xn}。
Preferably, the game payout of the zero sum game model is fi(xi),-fi(xi)。
Preferably, the solution of the first participant payment matrix satisfies:
Figure BDA0002797834060000025
in the formula (f)i(xj) Is an independent variable of xjThe optimal solution of the objective function in time.
Preferably, the mixing policy of the first participant satisfies:
Figure BDA0002797834060000026
θn＝1-θ1-θ2-…-θn-1，i＝n；
in the formula, alphaiAre coefficients, and the coefficients are obtained by converting the zero sum game model.
Preferably, the coefficients satisfy:
Figure BDA0002797834060000031
s.t.αi≥0；
Figure BDA0002797834060000032
preferably, the mapping in the step five is obtained by using the maximum value of the objective function and the optimized Pareto solution set in the same coordinate axis:
Ri＝θi(resultmax-resultmin)+resultmin；
in the formula, RiAs a coordinate axis-based solution, resultmaxFor the maximum value, result, of the optimized Pareto solution set in the coordinate axisminAnd collecting the minimum value in the coordinate axis for the optimized Pareto solution.
The invention has the following beneficial effects:
the zero-sum game-based multi-criterion decision method provided by the invention is characterized in that the zero-sum game thought is utilized, each objective function and a Pareto solution set after multi-objective optimization are determined according to a multi-objective optimization problem, the optimal solution of each single objective function is approximated to the solution set of the Pareto to enable the Pareto solution set to be optimal, the zero-sum game model reconstruction and the solution are carried out on the problem, the solution is a multi-criterion decision solution, the optimal multi-criterion decision method can be quickly found from the multi-objective optimization according to the Pareto solution set, the solution speed is high, the use is simple and convenient, and the solution precision is high.
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Fig. 1 is a schematic diagram of pareto solution set in the embodiment of the present invention.
Detailed Description
The present invention is further described in detail below with reference to the attached drawings so that those skilled in the art can implement the invention by referring to the description text.
The invention provides a multi-criterion decision method based on a zero sum game, which specifically comprises the following steps:
1. for a certain optimization problem, b sample points a are extracted by utilizing a Latin hypercube method1,a2,…,abAnd collecting response values o (a) corresponding to the sample points1),o(a2),…,o(ab)、p(a1),p(a2),…,p(ab) Etc. to build the approximate model required for optimization.
2. Aiming at a specific problem and combining an optimized approximate model, determining an optimized Pareto solution set and n objective functions of the problem, wherein each objective function is f1(x),f2(x),…,fn(x)；
3. Determining an optimal solution f for each objective function1(x1),f2(x2),…,fn(xn) The optimal solution is selected according to the following basis: approximating the optimal solution of each single objective function to a solution optimal in a Pareto solution set, wherein f1(x) Is f1(x1) The corresponding independent variable takes the value of x1；f2(x) Is f2(x2) The corresponding independent variable takes the value of x2；…；fn(x) Is fn(xn) The corresponding independent variable takes the value of xn；
4. Constructing a zero and game model:
1) game participants: a first participant (primary participant), a second participant (virtual participant);
2) and game strategy set: the strategy set of the first participant is each objective function fi∈{f1,f2,…,fnThe strategy set of the second participant is a set x of optimal solutions of all the objective functionsi∈{x1,x2,…,xn}；
3) And game payment: f. ofi(xi),-fi(xi)
4) The first participant's payment matrix is shown in table one:
table a first participant's payment matrix
Figure BDA0002797834060000041
Because the parameters are determined by adopting the hybrid Nash balance, and the payment policy of the first participant is each target function, the hybrid policy of the first participant is theta1，θ2，…，θn(ii) a Since the second participant is a virtual participant, the mixing policy of the second participant is defined as
Figure BDA0002797834060000042
Thus, the solution of the first participant payment matrix is:
Figure BDA0002797834060000043
where G is the solution of the first participant's payment matrix, fi(xj) Is an independent variable of xjOptimal solution of the objective function of time, thetaiFor the first participant's mixing strategy, i ═ 1,2, …, n,
Figure BDA0002797834060000051
j ═ 1,2, …, n for the second participant's hybrid strategy.
Therefore, the zero sum game model is as follows:
Figure BDA0002797834060000052
Figure BDA0002797834060000053
Figure BDA0002797834060000054
5. solving the zero sum game model:
wherein, the coefficient of the multi-criterion decision is determined by the mixed Nash equilibrium of the zero sum game, and for the convenience of solving, the game theory problem is converted into a linear programming problem, and the coefficient alpha can be solved by the following formula:
Figure BDA0002797834060000055
s.t.αi≥0；
Figure BDA0002797834060000056
the mixing policy for the first participant is then:
Figure BDA0002797834060000057
θn＝1-θ1-θ2-…-θn-1，i＝n；
6. calculating a mixing strategy element theta of the first participant under Nash balanceiAnd determining thetaiAnd mapping the value to an optimized Pareto solution set in a coordinate system according to the corresponding ith objective function.
The specific mapping method comprises the following steps: for a certain objective function, the maximum value of the Pareto solution set in the same coordinate axis is taken, and the solution based on the coordinate axis is as follows:
Ri＝θi(resultmax-resultmin)+resultmin；
in the formula, RiAs a coordinate axis-based solution, resultmaxFor the maximum value, result, of the optimized Pareto solution set in the coordinate axisminAnd collecting the minimum value in the coordinate axis for the optimized Pareto solution.
Solving R based on coordinate axisiThe vertical mapping to the Pareto solution set is the mostA multi-criteria decision method.
The invention designs and develops a multi-criterion decision method based on the zero-sum game, and by applying the thought of the zero-sum game, the optimal multi-criterion decision method can be quickly found from multi-target optimization according to a Pareto solution set.
Examples
The embodiment is in the automobile safety field, and the multi-objective lightweight optimization design of the aluminum alloy energy absorption box based on collision safety is carried out.
Full coverage of the variable range is achieved by the latin hypercube sampling method. Adopting Latin hypercube sampling method, setting sampling thickness interval to [0.5, 5%]The number of samples extracted was set to 30, and T (thickness), M (mass), E (energy absorption), F (energy absorption) were measured for each set of datamax(peak collision force) and faverThe response values (average collision force) and the calculated SEA (specific energy absorption) are specifically shown in table two:
values of the parameters of the two samples of the table
Figure BDA0002797834060000061
Figure BDA0002797834060000071
And establishing an approximate model required by optimization by using sample data obtained in the table II, wherein a multi-objective optimization design objective function of the problem is as follows:
Maximum SEA(x),faver(x)
Subject to Fmax(x)≤30kN
Figure BDA0002797834060000072
1mm≤x≤5mm
wherein SEA (x) is the specific absorption energy of the energy-absorbing box, faver(x) Is the average impact force of the section of the energy absorption box, Fmax(x) The maximum peak force of the energy absorption box in the collision process (Peak impact force), EbFor impact beam energy absorption value, EcThe energy absorption value of the energy absorption box is shown, and x is the wall thickness of the energy absorption box;
the objective function of the problem is to realize that the specific absorption energy of the energy absorption box is maximum with the average collision force of the section of the energy absorption box, namely: max SEA (x) and Max faver(x)。
The constraint function is that the maximum peak force of the energy absorption box does not exceed 30kN in the collision process, and the energy absorption value of the anti-collision beam is at least 30 times of the energy absorption value of the energy absorption box, namely:
Fmax(x)≤30kN；
Figure BDA0002797834060000081
the design variable of this problem is energy-absorbing box wall thickness, and the variable value range is: x is more than or equal to 1mm and less than or equal to 5mm, and a pareto solution set can be obtained through optimization as shown in figure 1.
Finding the optimal multi-criterion decision solution in the pareto solution set is solved:
1. finding the problem yields the objective functions:
objective function f1(x) The target function f is that the specific absorption energy of the energy absorption box is maximum2(x) The average collision force is maximum; the specific energy absorption of the energy absorption box of the objective function and the average collision force of the cross section of the energy absorption box are constructed by a Kriging model, and an explicit function formula is not used.
2. When the pareto solution set takes the optimal solution, f1(x) The maximum value is preferably 57.12, the corresponding solution x of its argument1＝4.5mm；
When the pareto solution set takes the optimal solution, f2(x) The maximum value is preferably 13.63, the corresponding solution x of its argument2＝2.3mm；
3. Constructing a zero and game model:
the game participant: a first participant, a second participant (virtual participant);
game strategy set: the strategy set of the first participant is each objective function fi∈{f1,f2The second participant }, the second participantThe strategy set of (a) is a set x of optimal solutions of each objective functioni∈{x1,x2}；
And (4) game payment: f. ofi(xi),-fi(xi)；
The payment matrix for the first participant is shown in table three:
table three first participant's payment matrix
Figure BDA0002797834060000082
Reconstructing the problem into a zero sum game model and converting the problem into a linear programming problem, wherein the solving process comprises the following steps:
maxα1+α2；
s.t.α1＞0
α2＞0；
57.12α1+13.59α2≤1
56.15α1+13.63α2≤1。
solving the linear programming problem of the above formula, the coefficients can be obtained:
α1＝0.0027；
α2＝0.0623；
the mixing policy for the first participant is then:
Figure BDA0002797834060000091
θ2＝1-θ1＝1-0.0102＝0.9898。
let theta equal to the maximum value theta2Taking the maximum value of the Pareto solution set in the coordinate axis:
resultmax＝13.63
resultmin＝13.59；
then the solution R based on the coordinate axis2Comprises the following steps:
R2＝θ2(resultmax-resultmin)+resultmin
＝0.9898*(13.63-13.59)+13.59
＝13.6296
so at this time f2Is 13.6296, f156.1597;
the optimal multi-criterion decision of the Pareto solution set is f1＝56.1597，f213.6296, the independent variable x is 2.33 mm.
While embodiments of the invention have been described above, it is not limited to the applications set forth in the description and the embodiments, which are fully applicable in various fields of endeavor to which the invention pertains, and further modifications may readily be made by those skilled in the art, it being understood that the invention is not limited to the details shown and described herein without departing from the general concept defined by the appended claims and their equivalents.

Claims (8)

1. A multi-criterion decision method based on a zero sum game is characterized by comprising the following steps:
the method comprises the following steps of firstly, extracting a plurality of sample points and collecting corresponding response values according to a multi-objective optimization problem, and establishing an approximate model;
determining an optimized Pareto solution set and n objective functions according to the approximate model;
step three, determining the optimal solution of the n objective functions;
step four, constructing a zero and game model:
Figure FDA0002797834050000011
Figure FDA0002797834050000012
Figure FDA0002797834050000013
in the formula, thetaiFor the first participant's mixing strategy, i ═ 1,2, …, n,
Figure FDA0002797834050000014
a blending policy for the second participant, j ═ 1,2, …, n, G, the solution to the first participant's payment matrix;
and step five, determining a corresponding objective function according to a mixing strategy of the first participant, and mapping the objective function to the optimized Pareto solution set in a coordinate system to obtain the optimal multi-criterion decision.
2. A zero-sum game based multi-criteria decision method as claimed in claim 1, wherein the zero-sum game comprises: the first participant and the second participant act as gaming participants.
3. The zero-sum game based multi-criterion decision method of claim 2, wherein the first participant's set of policies is an objective function fi∈{f1,f2,…,fnThe strategy set of the second participant is a set x of optimal solutions of the objective functioni∈{x1,x2,…,xn}。
4. The zero-sum game based multi-criteria decision method of claim 3, wherein the game payout of the zero-sum game model is fi(xi),-fi(xi)。
5. A zero-sum game based multi-criteria decision method according to claim 4, wherein the solution of the first participant payout matrix satisfies:
Figure FDA0002797834050000015
in the formula (f)i(xj) Is an independent variableIs xjThe optimal solution of the objective function in time.
6. The zero-sum game based multi-criterion decision method of claim 5, wherein the mixing strategy of the first participant satisfies:
Figure FDA0002797834050000021
θn＝1-θ1-θ2-…-θn-1，i＝n；
in the formula, alphaiAre coefficients, and the coefficients are obtained by converting the zero sum game model.
7. The zero-sum game based multi-criterion decision method of claim 6, wherein the coefficients satisfy:
Figure FDA0002797834050000022
s.t.αi≥0；
Figure FDA0002797834050000023
8. the zero-sum game based multi-criterion decision method according to claim 7, wherein the mapping in the fifth step is obtained by the least value of the objective function and the optimized Pareto solution set in the same coordinate axis:
Ri＝θi(resultmax-resultmin)+resultmin；
in the formula, RiAs a coordinate axis-based solution, resultmaxFor the maximum value, result, of the optimized Pareto solution set in the coordinate axisminAnd collecting the minimum value in the coordinate axis for the optimized Pareto solution.
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